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Chapter 4 More on Two-Variable Data YMS 4.1

Chapter 4 More on Two-Variable Data YMS 4.1. Transforming Relationships. Basics. Transforming data Changing the scale of measurement used when the data was collected Ch 4 Transforming Choose a power or logarithmic transformation that straightens the data

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Chapter 4 More on Two-Variable Data YMS 4.1

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  1. Chapter 4More on Two-Variable DataYMS 4.1 Transforming Relationships

  2. Basics • Transforming data • Changing the scale of measurement used when the data was collected • Ch 4 Transforming • Choose a power or logarithmic transformation that straightens the data • Why? We know how to analyze linear relationships! • Monotonic Function • f(t) moves in one direction as t increases

  3. Algebraic Properties of Logarithms • logbx = y if and only if by = x • Multiply/add • Log (AB) = Log A + Log B • Divide/subtract • Log (A/B) = Log A – Log B • Power to front • Log (x)A = A*Log x

  4. Growth • Linear • Increases by a fixed amount in each equal time period • Exponential • Increases by a fixed percentage of the previous total • y=abx

  5. Plot log y vs. x • If a variable grows exponentially, its logarithm grows linearly log y = log abx log y = log a + logbx log y = log a + xlogb

  6. Power Models • Ladder of Power Functions p201 • y = axp • Take logarithm of both sides straightens the data log y = log (axp) log y = log a + logxp log y = log a + plogx p213 #4.10-4.11 Homework: p222 #4.17 to 4.20

  7. YMS 4.2 Cautions about Correlation and Regression

  8. Some Vocabulary • Extrapolation • Predicting outside the domain of values of x used to obtain the line or curve • Lurking variable • Is not among the explanatory or response variables but can influence the interpretation of relationships among those variables • Can dramatically change the conclusions

  9. Reminders! • Correlation and regression only describe linear relationships and neither one is resistant! • Using averaged data • Correlations based on averages are usually too high when applied to individuals p230 #4.28 and 4.31

  10. Explaining Association • Causation • May not generalize to other settings • A direct causation is rarely the complete explanation • Is established by an experiment where lurking variables are controlled x y

  11. x y • Common Response • The observed association between x and y is explained by a lurking variable z • An association is created even though there may be no direct causal link z

  12. x y • Confounding • Two variables whose effects on a response variable are undistinguishable • May be either explanatory or lurking variables p237 #4.33 to 4.37 ? z

  13. Establishing Causation • Strength • There is a strong association between variables • Consistency • Many different studies show the same results • Response • Higher explanatory values produce a higher response • Temporal Relationship • Alleged cause precedes the effect in time • Coherence • The alleged cause is plausible/logical

  14. YMS 4.3 Relations in Categorical Data

  15. Two-Way Tables • Row variable/Column variable • Marginal Distributions • Found at the bottom or right margin • Are entire rows/columns over the total • Conditional Distributions • Only a cell that satisfies a certain condition (given in the row/column)

  16. Simpson’s Paradox • The reversal of the direction of a comparison or an association when data from several groups are combined to form a single group • Alaska Airlines vs. American West • Business vs. Law School Admissions Workshop Statistics 7-2 and 7-4

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